Overview

This document answers exercise questions from lesson 4 as well as explores the TS analysis framework provided by: https://otexts.com/fpp3/arima-r.html

Code Available on GitLab

Analyse Raw Data

Observed data

Transformations

The data is visually assessed to see if transformation or differencing is required. Additionally a KPSS test is performed in which the null hypothesis is that the data is stationary (small p-value -> differencing is requried).

Logging the data appears to do little in altering the shape of the data. Differencing appears to make the data more stationary. Also KPSS p-value of the orignal series is reported as .01 which indicates to reject the null. The KPSS p-value for the diffed series is reported as 0.1. Therefore diffed data will be used for model selection.

actual-data

## # A tibble: 1 × 3
##   county    kpss_stat kpss_pvalue
##   <chr>         <dbl>       <dbl>
## 1 Stockholm     0.775        0.01

log-data

## # A tibble: 1 × 3
##   county    kpss_stat kpss_pvalue
##   <chr>         <dbl>       <dbl>
## 1 Stockholm     0.776        0.01

diff-data

## # A tibble: 1 × 3
##   county    kpss_stat kpss_pvalue
##   <chr>         <dbl>       <dbl>
## 1 Stockholm     0.223         0.1

logdiff-data

## # A tibble: 1 × 3
##   county    kpss_stat kpss_pvalue
##   <chr>         <dbl>       <dbl>
## 1 Stockholm     0.166         0.1

Model Selection

The ACF appears to be significant at lag 1 and then cut off. The PACF appears to show geometric decay. I will choose an MA(1) model for the diffed series or ARIMA(0,1,1) on the original (ie the airline model). The automatic model was performed using the stepwise Hyndman-Khandakar algorithm. The algorithm chose an ARIMA(2,1,0) Which I would have never chosen by looking at the ACF and PACF.

Summary of manual model and automatic model:

## # A tibble: 2 × 9
##   county    .model   sigma2 log_lik   AIC  AICc   BIC ar_roots  ma_roots 
##   <chr>     <chr>     <dbl>   <dbl> <dbl> <dbl> <dbl> <list>    <list>   
## 1 Stockholm arima011  0.212  -12.5   29.0  29.7  31.0 <cpl [0]> <cpl [1]>
## 2 Stockholm stepwise  0.156   -8.74  25.5  28.1  29.5 <cpl [2]> <cpl [0]>

Manual

## Series: mort_rate 
## Model: ARIMA(0,1,1) 
## 
## Coefficients:
##           ma1
##       -0.4950
## s.e.   0.1509
## 
## sigma^2 estimated as 0.2118:  log likelihood=-12.49
## AIC=28.97   AICc=29.68   BIC=30.96

Auto

## Series: mort_rate 
## Model: ARIMA(2,1,0) w/ drift 
## 
## Coefficients:
##           ar1      ar2  constant
##       -0.9404  -0.5319   -0.2342
## s.e.   0.2191   0.2326    0.0856
## 
## sigma^2 estimated as 0.1559:  log likelihood=-8.74
## AIC=25.47   AICc=28.14   BIC=29.45

Forecast

Manual

Auto